Let $F$ be any field, $h:F^n\times F^n\to F$ symmetric bilinear form, $f:F^n\to F^n$ a linear transformation, let $M$ be a matrix of $f$ and $h$ in standard basis and $V\subset F^n$ a linear subspace. I have to show that $\dim f(V)+\dim V^\perp=n$.
I know how to prove this if $\det M \neq 0$ and if $\mathrm{char} F \neq 2$, when I can just find orthogonal basis of $F^n$, but I don't know how to show this in general case. I proved that if $x\in\ker f$ then $h(x, y)=0$ for all $y$ in $F^n$, so that all isotropic vectors are in $\ker f$, but i don't know if that's enough to find $\dim V^\perp$.